Imprint of the black hole singularity on thermal two-point functions

TL;DR

This work demonstrates that the high-frequency behavior of thermal two-point functions in holographic CFTs receives nonperturbative corrections governed by null geodesics that bounce off the black hole singularity. Using a bulk WKB/steepest-descent approach, the authors show these corrections are encoded in a reflection coefficient that connects near-singularity physics to the exterior boundary via a transseries in the frequency, $G_{ m ret}( om{ω})= om{ω}^{2 u}[ ext{leading perturbative term} + i e^{-rac{eta om{ω}}{2}(1- ext{i})} ext{R}( om{ω})+ dots]$, with $ u= riangle-rac{d}{2}$; in particular, $R( om{ω})=-2+O( om{ω}^{-2/3})$ for scalar probes and the leading nonperturbative piece corresponds to two bouncing geodesics (one forward, one backward in time). The paper provides explicit OPE-based predictions for the large-$ om{ω}$ expansion, derives a bulk-boundary matching that yields a canonical Borel resummation of the perturbative series, and presents numerical tests (e.g., for Δ=4 in d=4) that confirm the predicted coefficients and large-order behavior. Overall, the work strengthens the link between interior black hole dynamics and exterior observables, suggesting new avenues to probe and reconstruct the interior geometry from boundary data.

Abstract

We consider two-point functions of light fields at finite temperature and large real frequencies in holographic theories. The thermal system is dual to a single-sided AdS black hole. We show that the high-frequency expansion obtained from the Operator Product Expansion receives nonperturbative corrections, which are controlled by null geodesics bouncing off the black hole singularity in the two-sided eternal black hole geometry. We develop a bulk WKB description of these bouncing geodesics and explain how to calculate reflection coefficients at the singularity.

Figures (6)

• Figure 1: Current spectral density \ref{['chi']} at finite and zero temperature. We will focus on the exponentially decaying difference at large frequencies; the dips are caused by destructive interference between the two paths in Fig. \ref{['fig:geod1']}.
• Figure 2: Nonperturbative contributions $\sim e^{-\frac{1}{2}\beta\omega+\mathrm{i} \omega t_1}$ to the retarded function at high frequencies will be explained from a null geodesic that reflect once off future singularity of the eternal black hole. Wightman functions and ${\rm Im}\, G_{\rm ret}$ also receive contributions from the time-reversed geodesic.
• Figure 3: The original real-time contour (in red) for the Fourier transform of $G_{\rm ret}(t)$ can be deformed into a steepest-descent contour along the imaginary axis plus a branch cut starting at $t = \frac{\beta}{2} (1 + \mathrm{i})$. The same cut appears in the Wightman functions $G^>(t)$ continued to the second sheet.
• Figure 4: Stoke's lines (in blue, with phases $n\pi/3$ near the origin) where WKB solutions exchange dominance, and lines of constant WKB phase (solid red and gray dashed). We track $\phi^{\rm steepest}(r)$ from $r=-\mathrm{i} r_h$ to large $r$ following the solid path with arrows.
• Figure 5: The spectral density $\rho=2{\rm Im} G_{\rm ret}(\omega,q=0)$ for a scalar with $\Delta=4$, showing the decreasing residuals after subtracting various numbers of terms in \ref{['large omega ansatz']}. The dashed line is $4\omega^4e^{-\frac{\beta\omega}{2}}$.